The given problem asks you to find the limit:

$\lim_{x \to a} \frac{\tan(x) - \tan(a)}{x - a}$

This expression resembles the definition of the derivative of a function at a point. Specifically, this is the derivative of $\tan(x)$ at $x = a$. The derivative of $\tan(x)$ with respect to $x$ is:

$\frac{d}{dx} [\tan(x)] = \sec^2(x)$

Thus, applying this to the limit:

$\lim_{x \to a} \frac{\tan(x) - \tan(a)}{x - a} = \sec^2(a)$

So the value of the limit is:

$\sec^2(a)$

Would you like a more detailed explanation of the derivative approach?

Here are 5 related questions to expand on this topic:

1. What is the definition of a derivative using limits?
2. How do we differentiate trigonometric functions like $\sin(x)$ and $\cos(x)$?
3. What is the derivative of $\tan(x)$ and how is it derived?
4. Can this type of limit be solved using L'Hopital's Rule? If so, how?
5. How would you calculate the limit if $x \to 0$ instead of $x \to a$?

Tip: When working with limits involving trigonometric functions, knowing the derivatives of these functions can simplify the process tremendously.